Two lines with a third intersecting line

Types of Angle Relationships and Angle Relationship Names
There are six types of angle relationships. Here is a table showing the types of angle relationships and the angle relationships' names. Use the aboveangle picture for reference.
Angle Relationship 
Example 
Corresponding Angles 
Angles a and e 
Alternate interior angles 
Angles d and e 
Alternate exterior angles 
Angles b and g 
Consecutive interior angles 
Angles d and f 
Consecutive exterior angles 
Angles b and h 
Vertical angles 
Angles e and h 
The two alternate angles are grouped under alternate angles and the two consecutive angles are grouped under consecutive angles. The interior and exterior designation describes which pair of alternate or consecutive angles are being referred to.
Alternate and Corresponding Angles
Alternate and corresponding angles are formed when two lines are intersected by a third transversal line. Alternate angles are those angles that are on opposite sides of the transversal (the blue line in the reference picture above). Corresponding angles are those angles that are in the same relative positions at each intersection.
There are two types of alternate angles including alternate interior angles and alternate exterior angles. Both will be discussed in more detail.
Alternate Interior Angles
Alternate interior angles are the angles between the two lines being intersected and on opposite sides of the transversal. Using the reference picture, there are two pairs of alternate interior angles.
Alternate interior angles

The green angles c and f form one pair of alternate interior angles and the purple angles d and e form the second pair of alternate interior angles. Both pairs are inside the two black lines and each angle is on the opposite side of the blue transversal.
Alternate Exterior Angles
Alternate exterior angles are alternate angles that are outside the two lines being intersected by the transversal. Just like there are two pairs of alternate interior angles, there are also two pairs of alternate exterior angles.
Alternate exterior angles

One pair of alternate exterior angles are the green angles a and h. Another pair of alternate exterior angles are the purple angles b and g. Both pairs are outside of the two black lines with each angle being on the opposite side of the blue transversal.
Corresponding Angles: What Do Corresponding Angles Look Like?
Corresponding angles are the angles that are in the same location at each intersection. For example, if one angle is at the NW side of an intersection, then a corresponding angle will also be at the NW side of another intersection. So what do corresponding angles look like?
Corresponding angles

The pair of red angles a and e are corresponding angles. Notice how both are in the same location at their respective intersections. Both are at the top left or NW part of the intersection.
An intersection has four sections, so that means there are four pairs of corresponding angles. The other corresponding angles are angles b and f, d and h, and c and g. Notice how they all are located at the same relative positions at their respective intersections.
Are There Corresponding Interior Angles and Corresponding Exterior Angles?
The question as to whether there are corresponding interior and exterior angles is a common question. This is probably due to the fact that there are alternate interior and exterior angles. But for corresponding angles, this is not the case. Note how there are four corresponding angle pairs where there are only two alternate interior angle pairs and only two alternate exterior angle pairs.
Corresponding angles cannot be all interior nor can they be all exterior. Because they are relative to the location at an intersection, one angle will always be interior while the other angle will always be exterior. If both angles are interior, then they cannot be corresponding since their relative position at an intersection has changed.
Consecutive Interior Angles
Consecutive interior angles are interior angles that are on the same side of the transversal. These angles are not on opposite sides of the transversal.
Consecutive interior angles

The orange angles d and f make up a consecutive interior angle pair. Both are interior angles being inside of the two black lines and both are on the same side of the blue transversal. There is one more pair of consecutive interior angles. The angles c and e make up that other pair. Just like there are two sets of alternate interior angles, there are two sets of consecutive interior angles.
Consecutive Exterior Angles
Consecutive exterior angles are exterior angles that are on the same side of the transversal.
Consecutive exterior angles

The orange angles a and g make up a consecutive exterior angle pair. Just like there are two pairs of consecutive interior angles, there are also two pairs of consecutive exterior angles. The other pair are the angles b and h. Both these angle pairs are outside the two black lines and both pairs are on the same side of the blue transversal.
Vertical Angles
Vertical angles are the angles that are opposite each other at an intersection. Standing at an intersection, the vertical angle is the one that is diagonally on the other side. In most city intersections, there is no diagonal sidewalk. So it takes crossing one street and then the other to get to the vertical angle. Vertical angles do not require three lines. Every single intersection of two lines forms two pairs of vertical angles.
Vertical angles

At this intersection, the orange angles make one vertical pair and the purple angles make another vertical pair. Note they are opposite each other at the intersection or diagonal to each other when standing at an intersection. Vertical angles share the vertex but do not share any sides.
When are Angles Congruent?
Angles are congruent when they have the same measurement. For example, if two angles both measure 63 degrees, then they are congruent. But if one angle measures 63 degrees and the other measures 64 degrees, then they are not congruent.
The abovementioned angle relationships are sometimes congruent under special circumstances. The special circumstances are discussed below.
Vertical Angles
Vertical angles are always congruent. This is because vertical angles are essentially the same angle just on the other side of the line. Lines are also called straight angles because they always form a 180degree angle. When a line is intersected, the two consecutive angles are always supplementary. So in an intersection, if the angle to the right and left of the angle is supplementary to that angle, then the vertical angle must equal to that angle since the angles to the left and right of the vertical angle are supplementary as well.
Are Corresponding Angles Always Congruent?
Corresponding angles are not always congruent. The special circumstance where corresponding angles are always congruent is when the two lines being intersected by the transversal are parallel. When this happens, the transversal creates two identical intersections and, therefore, each corresponding angle at each intersection is congruent to its corresponding angle partner.
Are Alternate Angles Congruent?
Alternate angles are congruent only when the two lines being intersected by a transversal are parallel. When this is the case, the two intersections are identical and the alternate angles essentially form a vertical angle pair. This applies to both alternate interior angles and alternate exterior angles.
Are Consecutive Interior Angles Congruent?
Consecutive interior angles are congruent only when the transversal intersects two parallel lines at 90 degrees, meaning the transverse is perpendicular to the two parallel lines. When this happens, all the angles are 90 degrees and therefore the consecutive interior angles are congruent with both being 90 degrees. If the transversal is not perpendicular, but the lines are parallel, then consecutive interior angles are supplementary.
Are Consecutive Exterior Angles Congruent?
Consecutive exterior angles are congruent under the same circumstances as consecutive interior angles. If the transversal intersects the two parallel lines at a perpendicular angle, then the angles are congruent. If the transversal intersects two parallel lines but is not perpendicular, then the consecutive exterior angles are supplementary.
Lesson Summary
In review, an angle is defined as the amount of turn between two lines, segments, or rays. Common types of angles include acute, obtuse, right, straight, and reflex angles. Alternate interior angles are the angles between the two lines being intersected and on opposite sides of the transversal. Alternate exterior angles are alternate angles that are outside the two lines being intersected by the transversal. Corresponding angles are the angles that are in the same location at each intersection. Consecutive interior angles are interior angles that are on the same side of the transversal. Consecutive exterior angles are exterior angles that are on the same side of the transversal. Vertical angles are the angles that are opposite each other at an intersection.
Vertical angles are always congruent. Alternate and corresponding angle pairs are congruent when a transversal intersects two parallel lines. Consecutive angles are congruent only when a transversal intersects two parallel lines perpendicularly. Otherwise, consecutive angles are supplementary when a transversal intersects two parallel lines.